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No, any number of Infantry Divisions can carry any number of nukes. Here are some examples of legal commands:

INF + SRNUKE @ B1 > BRAXX
(One INF carrying one SRNUKE from B1 to BRAXX)

INF*4 + SRNUKE @ B1 > BRAXX
(Four INF carrying one SRNUKE from B1 to BRAXX)

INF*4 + SRNUKE*2 @ B1 > BRAXX
(Four INF carrying two SRNUKE from B1 to BRAXX)

INF*4 + SRNUKE*2, MRNUKE @ B1 > BRAXX
(Four INF carrying two SRNUKE and a MRNUKE from B1 to BRAXX)

As one command, two INF can carry two nukes. An example of such a command is:

INF*2 + SRNUKE*2 @ B1 > BRAXX

In the same round, the nukes may also be fired, but a separate command would be needed for each strike:

S: SRNUKE @ BRAXX > B6

And (assuming the player has a Command Center providing an additional command):

S: SRNUKE @ BRAXX > B5

Moves and move resolutions take place before strikes, so your Armored Divisions will "fight" after moves, regardless of any strikes that are taking place. As an example, suppose you are moving your ARMOR from B1 to BRAXX, and then intend to strike an Industry Complex in B6 in the same round. Your commands would look like:

M: ARMOR @ B1 > BRAXX
S: ARMOR @ BRAXX > IC @ B6

Now, if there is a conflict (such as another player moving ARMOR to BRAXX) whereby your Armored Division is destroyed, then your strike will simply fail (on account of not having ARMOR in BRAXX to strike with). On the other hand, if your ARMOR is still standing after move resolutions, then your strike will succeed. Note that strikes generally take place at the same time, so even if BRAXX were nuked, and your ARMOR destroyed, it would still get off its strike on the Industry Complex.

Yes, assuming that the second hex is adjacent to the one the ARMOR is moving to. Your commands might look something like:

M: ARMOR @ B1 > BRAXX
S: ARMOR @ BRAXX > IC @ B6

In the applet game board, it’s easiest to construct the strike first by dragging the targeting crosshair from the Armored Division where it currently sits (at B1) to the target, whereupon your command would look like:

S: ARMOR @ B1 > IC @ B6

Then, hold down the shift key and click BRAXX, and your command will change to:

S: ARMOR @ BRAXX > IC @ B6

You CAN move a transport without any INF, if you occupy the hex it’s in to begin with, but if you send it to an empty hex, you’ll lose control of it.

Patriot missiles will NOT counter-strike against Jets. They used to take out Bombers, but the catalog has been changed so that they no longer do. A Patriot will automatically counter-strike a nuclear missile before a neutron missile. This targeting priority is based on a generalization of expected damages.